336 



REPORT — 1869, 



Section 2. — In the eighth volume of the ' Cambridge and Dublin ATathema- 

 tical Journal ' there is a paper by Mr. F. JS^ewman " On the Third Elliptic 

 Integral.'' The i^rincipal object of this paper is to apply Lagrange's trans- 

 formation to this integral so as to obtain available results. The interest and 

 difficulty attaching to this are great, and I think it well, therefore, to pre- 

 sent Mr. j^ewmau's leading theorem to the reader in a form which he will 

 find it easy to follow. 



Let 



Ai = V i —c^' sin- d^, A = V 1 — c^ sin= d, 



U(c, n, d)= \7T—, • o „s ■ » 



■ ^ ' ' -^ ](l + nsin-'y)A 



T(c,n, e)=n(t% », 6)- 



F(c) 



n(c', n). 



Then, following the usual scale of Lagrange, if 



■a /I , ,, sine cos , /T— ^ 

 sm0^-(l + () , ,. = Vi_r, 



we have 



1-e' 



J(H-MjSiii^0jA, ^ '^^jA=-|-«,(l + c7sin=flcos^0 ' ' ' ^^ 

 Let us assume 



. .A^4-«j(l + Osiji'^cos-0 = (l + «sin=0)(l— msin^e), 

 where tn and —n are two new parameters. Then 



nm=n^{l + cy\ {l + n)(l-m) = c'\ 



1 



r' 



J 



whence, since remembering the second of equations (2), 



A _ 



(l-)-yi sin- 0)(1 — w sin- f)~ 



■ ■ jnm^ I 1-l-w 1 1— TO 



m + n\ n ' (l+Msin-fl)A m ' (T 



Therefore from (1), 



]](<■„ «,o,):.(i+O-^jl±^n(<v,,0)+L-z^'n(<-,-,., 0)1, 



m-\-n { n ^ ^ m ^ ^ J 



It 

 putting 0= 2' 'I"*! therefore 0^ = 7r, 



2n(v«,)=0+0-!!i!L|I±^n(.,»)-t-i^'n(r,-»o). 



m + n { n m J 



(2) 

 (3) 



_J 1 



—TO sin'' 6) A J 



